The sum of two numbers is $83$, and their difference is $43$. What are the two numbers?
Explanation: Let $x$ be the first number, and let $y$ be the second number. The system of equations is: ${x+y = 83}$ ${x-y = 43}$ Solve for $x$ and $y$ using elimination. Add the top and bottom equations together. $ 2x = 126 $ $ x = \dfrac{126}{2} $ ${x = 63}$ Now that you know ${x = 63}$ , plug it back into $ {x+y = 83}$ to find $y$ ${(63)}{ + y = 83}$ ${y = 20}$ You can also plug ${x = 63}$ into $ {x-y = 43}$ and get the same answer for $y$ ${(63)}{ - y = 43}$ ${y = 20}$ Therefore, the larger number is $63$, and the smaller number is $20$.